It turns out there is one thing that everyone knows about engineers: We Do Maths. I’ve lost count of the amount of times friends have simply deferred any arithmetic to my good-self because “you’re an engineer- you do maths“; dividing bills, working out petrol costs and calculating quantities- my apparent skills have no end. It’s got to the point where my somewhat-taller-half no longer actually does any maths, he just wonders out loud any arithmetic he requires the answer to and then stares meaningfully at me until I answer.

The slightly worrying thing about this whole situation is that I’ve also lost count of the number of times I’ve witnessed an engineer (myself included) struggle with a trivial calculation, joking: “it’s lucky I don’t do maths for a living…”. Of course, the fact is that the sort of maths engineers use on a daily basis has a habit of being a little more abstract and calculator based than your day-to-day restaurant disagreement. It doesn’t hurt, however, to know a few of the following tricks to make ball-parking easier and maintain the slightly dubious reputation engineers have for “doing maths”.

Double and Halve

People are good with the number two; whether it’s because it’s the first times-table most of us learn, or just because doubling and halving quantities comes naturally to us. By appreciating that we can almost instinctively divide and multiply by two you can substantially improve your mental arithmetic speed; it’s called the double-and-halve trick. When multiplying two numbers just halve the smallest and double the largest until you end up closer to a factor you’re familiar with:

Obviously this trick is at its most effective when you have a number that is a power of two. For those times when life gives you lemons, however, you can still achieve similar speeds by separating out the difficult portion:

Form the Square

Squaring numbers is a common thing in engineering; probably because of they way we work with physical quantities such as area. This trick is actually the application of a simple rule, and an equation that everyone learns and immediately forgets. For those of you who don’t know it, there is a rule for squaring any number ending in five: form a number using all but the last digit (i.e. 5), multiply that by itself plus one, and then finish it with 25:

It even works with decimals, just temporarily remove the point by multiplying though tens:

Of course, you’d be lucky if you managed to pull off a calculation where you only had to square something ending in a five. However, you are only ever ±5 away from a number ending in 5, and that’s where our GCSE maths equation makes an appearance:

Using this little gem, we can split our non-conforming number into something ending in five, and a number between -5 and 5 that we should be able to square in our heads. This allows us to use the ‘ending-in-five’ trick even when the number doesn’t:

Or, similarly, where you nearest five is greater than the number you’re trying to square:

Engineers Aren’t Mathematicians

This final trick isn’t really a trick at all, it is just the recognition that engineers aren’t mathematicians; we operate in uncertainties and precisions. Whilst I can’t calculate the square root of 73 in my head, I know that the answer is between 8 and 9, and is closer to 9; at a guess I’d say [Ed. you’ll have to trust me on this one:] 8.6, which (at 73.96) is good enough the majority of Civil Engineering applications.

By keeping an eye on the degree of accuracy you’re simplifying each stage by, you should be able ensure that you can calculate to a good enough aim for the application you need. You can improve your accuracy by acknowledging that, as an engineer, you’re trained to think conservatively and therefore you will want to normalise your conservatisms by taking geometric means between ranges (see Fermi Estimates for a guide).

It should be noted that being ‘close-enough’ when dealing in bistromathics is, alas, never good enough!

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Over the last year I’ve been learning some real practical engineering lessons from the most unlikely of sources: my IKEA Austmarka wardrobe. Brought from that nightmarish trip, which has become the modern day ritual for all people moving into an unfurnished place; I estimate that this modular masterpiece survived for about three months before it needed some engineering assistance.

My engineering lesson came at two in the morning when Austmarka decided that supporting the clothes rail was something of an optional duty. After a week of playing ‘if I just add one more shirt will the whole rail come down’-buckaroo, one of the supporting clips sheared-off. It was at this point that I began my life as a forensic engineer.

BIM Level 7.

As my Leonardo’ific sketches show, the support for the clothes rail is a plastic cup that slots, using two shear-pins, into the wardrobe wall. Making the assumption that I do not store my anti-gravity clothing in this cupboard, the clothes rail is propped vertically by the cup, and restrained from making a lateral escape by the wall itself. The weight of the rail (and the hanging clothing) is then transferred through shear by the two pins, which sit in small holes along the wardrobe wall.

The connection detail makes an assumption; there is no moment. This isn’t strictly true, and is the point where it all begins to fall down (literally). Our ‘cup’ is essentially a bearing-plinth/pad-stone, and, as a (very) quick rule-of thumb, this means a semi-triangular distribution of load with the equivalent ‘point’ vertical force being about two-thirds along the triangle. That means there is eccentricity; and with eccentricity comes moments.

Just to make things worse for us, the tolerances on the rail are surprisingly high; if the rail is flush with one side, the length only extends about half-way along the other support. A little bit of statics shows us that the moment increases the shear through the pin connections by the ratio of effective rail eccentricity to about a-half to one-thirds of the pin-length. In our case that more than trebles the shear through the pin; which is beyond even a civil engineering factor of safety.

Failing around the letter P.

Actually, the above statement is a bit of a conservative simplification, because the top pin would add additional restraint. This complication, however, never arose, as the tolerance around the top-pin’s hole was enough to let it beat a hasty retreat from the socket, out into a world free of constraint. This left our bottom pin to deal with exactly that conservative simplification on its own; at which point, it gave up on life and sheared straight through.

Whilst I would still suggest the more obvious solutions of having IKEA design taking into account the worst-case geometric effects of their tolerances; something that irked me about the detailing of the support was that the second pin could have been prevented from disengaging so easily by simply placing the cup at the top, instead of the bottom. The bar would then act as a prop, allowing the increased shear to have at least been shared across both pins.

For anyone that came here looking to fix their wardrobe, we’ve found some good mileage in using 1p coins as shims to ensure the clothes bar short-tolerance is distributed evenly between both supports, and then nailing through the top-pin to hold the assembly in place. Possibly there is something in using steel rings, or modelling clay, to create a distribution plate/pad and reduce the effective eccentricity of the rail; but life is a little short.

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This week I was randomly introduced to the wonderful world of Fermi problems. If, like me, you have no idea what I’m going on about: Enrico Fermi was a physicist who was renowned for making good approximations with no actual data, and a Fermi estimate is a method for making a ball-park guess that might actually be right (for a given value of right). I know what you’re thinking: but that’s just engineering design!

Actually, it gets even eerier. The basis of the method is to divide a problem into several factors that you are better positioned to estimate, and combine them to provide a figure with little or no technical calculation. It’s starting to sound more and more like preliminary design- no? For example:

• There are 8 million people living in London
  
• The Shard took 3 years to build
• The magnitude of people working, given any day, on the Shard must have been in the hundreds… at an estimate I’d say between 50 – 250…
• … in the case of a range, take the geometric mean (the square root of the product)…
• … but because using a calculator is against the spirit of the method, estimate the geometric mean at the lower third: 50 + (250 – 50) / 3 is approximately 120.
• There are 360 days in a simple year, so it took 1080 days for 120 people to build the Shard…
• … or somewhere close to 1100 days for 100 people to build it for easy multiplication: 110’000 people days …
• … which, to the nearest magnitude, is a tenth of a million.
• So if all 8 million people in London stopped being so damn lazy and chipped in to that tenth of a million people days …
• … it would have taken a (1 / [8 / 0.1]) 80th of a day …
• … and as 24 hours fits almost trice into 80; that’s about a third of an hour …
• Which is about 20 minutes; but for the degree of accuracy:
• Let’s say a day.

As an example, this one is a bit more deliberate than it first seems. As a process, I expect most people who have ever done preliminary design recognise the method; even if you don’t give Fermi the credit: Idealise, simplify and state your assumptions. Similarly, the accuracy of the calculation aims for degrees of magnitude; whilst you might not know how many years the Shard took to build, you could guess it is a single figure.

Something I hadn’t been introduced to, before I started looking at the method, was use of a geometric mean to reduce a range to a representative value. I think there is a tendency in engineering to conservatism, which can cause a significant over-estimate. The geometric mean therefore helps us on two levels, at once it removes the temptation to play it safe with the highest value, and, because it tends towards the smaller number, allows us to acknowledge that our lower bound would have been conservatively picked, and the upper bound is most definitely exaggerated.

My final step of moving from 20 minutes to a day is a reflection of this. Because of the way the assumptions are multiplied, working always with conservative estimates compounds the conservatism of the design until your estimate is so far away from likely that you might as well have just guessed at the start. As the use of the method is most helpful prior to any detailed design, however, the end is the best place to apply a safety factor.

An advantage of using a formal Fermi method is that this estimate can provide a basis of sanity for further calculations. When a detailed analysis is undertaken, answers not within the magnitude of the estimate require the identification of the assumptions that led it out of kilter; justifying the sanity of the detailed calculation.

Finally, I’m sure anyone with any site experience might argue that, even if you could handle the logistics, eight million people could not build the Shard in a day, because actions such as curing, lifting, boring, etc. all impose time constraints. This highlights a limitation of the method; it requires linearity, or at least a scalar between the basis upon which the assumptions were made and the answer. Whilst there will be a range that throwing more people at the construction would have got it built faster, somewhere between that and a day new, non-modelled, effects begin to dominate.

Despite all this, an awareness between the similarities of the Fermi method and preliminary design calculations is useful. Appreciating the analogy helps both to formalise the process, prevent our tendency from conservatism snow-balling into a significant over-design, and leaves us with a set of assumptions to hone and evaluate when it comes to the details.

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I expect I’m not alone in wishing for a day where my design technique is so perfect I can just expel completed structures from my pencil onto the calculation pad. Until then, however, a major factor of any design is iteration; assumptions are proven wrong, the site investigation returns unexpected results, initial member sizes are under capacity… It all ends in revisiting and reiterating. This would be fine if design was separate from construction, but often program and pre-costing requirements fix the design before it is truly completed, and it’s easy to find yourself stuck into a corner. This is especially true for retro-fitted works, where an existing structure doesn’t quite make the grade, but the client can’t afford a new one.

Here are a few techniques I’ve picked up over my time, which can offer an escape when things just aren’t working out.

Common Sense

Beware of stair climbing fire trucks…

There’s always the “but would it really happen?” discussion. Engineering codes tend to be comprehensive, focusing on major Civils works and the many possible actions that might be-fall them. As a rule, standards should be followed, but when the design is tight it may be time to ask yourself if an over-weight articulated lorry really can get up the tiny access ramp? My favourite example of this is the new ‘service vehicle’ load (which weighs as much as a fire-engine) defined by EC1 for foot-ways, but if you’ve got a pedestrian footbridge with stairs on both sides, it’s probably safe (with client consent) to discard it, unless there’s likely to be significant changes to the access in the future.

Moment Redistribution

Moment redistribution is a hard phenomenon to explain to the uninitiated (lit. someone who can say the phrase full-penetration butt-weld without laughing…). Essentially most construction materials do not fail immediately, but enter a stage of permanent deformation. At the point of deformation the material becomes weaker, which is where moment re-distribution comes in to play. Loads are attracted to stiffness like a bad taste simile to an inappropriate joke, and so the peak load that caused the local failure begins to spread, and the high moments start to become more forgiving. For the likes of reinforced concrete this can mean a more involved analysis, but it’s often the way to a more economic solution.

Modelling Quirk

I once heard someone describe models as a complex way of repeating what you told the computer. This is somewhat true, and quite often the unexpected failure in the structure is simply a quirk of the model. Typically if an effect can’t be replicated by a hand calculation, there is good cause to be dubious. For example, Finite Element models have a tendency to develop peak stresses, which would normally result in relieving deflections: Make a judgement.

Stress History

With complex loading scenarios it can often be worth consider “does it all happen at once?” For some materials (like pre-stressed concrete) this aspect is included as part of the normal design method, however where two structures are connected stress established before the interface (typically dead-loads) will not be transferred. By extension another consideration is “will this make it worse?” because, if a structure is well-established, works that change, but do not worsen, the loading conditions are unlikely to have a detrimental effect on the structure (although it’s a completely different story for geotechnical works!)

Reducing the Factor of Safety

A depressingly true story.

Factors of Safety, when it comes down to it, are just a formal mechanism for over-designing structures to provide capacity against the unknown. By understanding what each partial factor is protecting against, and taking preventative action at another stage, you can begin the justify a reduction. Typically this needs stakeholder buy-in, but its often a way to avoid excessive repairs to existing structures. Examples include reducing the factor on surfacing by specifying the surface to be planed before re-laying, thus preventing the gradual material build-up the factor emulates.

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It is surprisingly easy, sitting in the office, to forget that the bunch of design calculations and drawings you’ve been pondering over all morning will one day have to be built by someone. Consideration of constructibility is a CDM requirement nowadays, and the dream of designing something that doesn’t cause a contractor to baulk is not yet dead.

So here are five, of the many, deadly sins that consultants often commit, which cause contractors to die a little inside (so I’m told)…

Rebar for Superman

Although no longer health and safety cannon the ’25 kg’ rule is still a good way of gauging what an individual can lift. This means that using 40 mm bars should be avoided, unless you’re planning to employ a gang of weight-lifters to build it.

Avoiding the largest rebar is only the start, however. The following table gives the weight of the maximum 12m length bar, and the length of a 25 kg bar for each diameter. Note that you need a small camel to lift a full H40 length.

Amazon Wish Lists

Not everything on a civils project can be designed using the fundamental elements of concrete and steel. Eventually you’re going to need some proprietary assistance, and the time comes to start specifying products. Often this selection will have some influence on the design and, if you’re lucky, you’ll be able to ask the contractor for a preference. More often than not, though, you’ll have to make the first guess.

In these situations, I have found, its better to consider product selections as assumptions. To be competitive contractors have to be able to shop around, and (unlike a lot of architects) suppliers don’t pay commission to designers. The problem is that the contractor cannot safely just pick anything. For example, the breaking load of steel parapets vary wildly between manufacturers; the wrong product could cause the structure to fail.

Whatever product is selected needs to meet both the specifications of, and the assumptions behind, the design. The following almighty three words, however, are our saviour of compromise: ‘Or Similar Approved‘. Tagging this note onto any named product gives the contractor scope to find the best deals (as most suppliers can provide similar products to their competitors) while still ensuring the consultant gets a chance to check it against the design.

Digging to the Centre of the Earth

A few years ago the rules for excavations were fairly simple, and trenches shallower than 1.25m didn’t need support. Now it’s much more complicated, and digs require support if they pose a risk. Regardless of this, many engineers still start with a 1.25m is safe attitude; and then work from there.

Digging, especially around buried services, is a difficult and laborious task. It follows that with almost every drainage or services design I’ve been involved with, the contractor has always asked ‘can we make it shallower?’.

There’s no ‘one-size-fits-all’ solution here, but with trench like excavations you can consider using oversized pipes to lessen long gradients- or use engineered, rather than nominal, cover requirements to bring buried aspects of the design back up to ground level.

Equal Opportunity Design

On the surface, the existence of the term ‘Contractor Designed Detail’ is a little bewildering. This is especially true when you consider that the basic definition of a engineering consultant is someone who designs. Of course there are times when you simply cannot design something until the contractor is involved. Typically this is when the project requires specific construction methods, or employs specialist techniques that require input from sub-contractors.

But with great power comes great responsibility, and abuse of this note will win you no friends.

The Ikea Instruction Manual

Every designer should aim to create economical engineering solutions, working with high utilisations and no wasted parts. There comes a point, however, where the finish product becomes too meticulous to be reasonably built.

Creating connections, for example, made out of seven different bolt sizes for maximum efficiency is, at best, likely to come back with a plea for rationalisation or, at worst, result in some serious construction mistakes.

Providing a single, worst case, pile design and repeating it for hundreds of piles will also, however, rub your contractor up the wrong way; contemplating all the money lost on a truly uneconomical design.

The art is to aim for enough division to provide the perfect balance between economy and buildability; which is about as close as Civil Engineering gets to Zen Philosophy.

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Not so long ago the aim of structural analysis was to develop simple, elegant, and reliable methods of design. The idea of splitting a member into thousands of tiny elements and calculating their individual behaviours would have been laughed at. Of course now finite element analysis is common place, and the thought of designing a complete project without using a computer is unheard of.

The design tables and empirical rules devised before, however, are not without merit. Although no longer accepted design practice, they provide a valuable tool for getting the ‘feel’ of a design way before the detailed calculations have been finished

So, here are five of the many odd ‘engineering rules of thumb’ I’ve picked up and adopted over time.

Instant Bridge Design

Integral reinforced concrete bridges are becoming more and more prolific. Given their potential for low maintenance (no joints) they are now preferred both by the Highways Agency and Network Rail.

In 2006 Braun researched the typical span to depth ratios for integral reinforced concrete bridge designs. Using these findings its possible to sketch a bridge from the span alone.

Typical span/depth ratios for highway bridges:

Typical span/depth ratios for rail bridges:

Statics at a Glance

Here are two short-cuts for static analysis:

• The end rotation of a well designed, simply supported beam is 0.2^c.
• Splices in continuous beams should be at the quarter points.

Bricking It

The following height to thickness ratios can be used to find a starting size for masonry walls:

• Solid Bearing Walls: 1:20
• Bearing Walls with gaps: 1:18
• Non-bearing exterior walls: 1:18
• Non-bearing interior walls:1: 36

One Line Floor Slab

A good estimate for the depth of a concrete floor slab is the same as a 1/180 of its unsupported perimeter.

Revealing Diameters

The flow through a pipe is roughly 40% of its diameter, and the minimum fall should be 1: the pipe diameter to ensure self-cleaning velocities.

Of course all of these engineering rules of thumb are just taken from experience and observation, and every element should be properly designed before construction. But when it comes picking a starting point, a few golden rules go a long way.

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